# Divide and conquer method for proving gaps of frustration free Hamiltonians

## TL;DR

This paper introduces a method to establish system-size independent lower bounds on the spectral gap of frustration free Hamiltonians, linking ground state properties to spectral gap estimates across dimensions.

## Contribution

It presents a novel approach connecting ground state space properties to spectral gap bounds, providing necessary and sufficient conditions for a constant spectral gap.

## Key findings

- Ground state space property suffices for spectral gap bounds
- Necessary and sufficient condition for constant spectral gap
- Upper bound on spectral gap for gapless models in any dimension

## Abstract

Providing system-size independent lower bounds on the spectral gap of local Hamiltonian is in general a hard problem. For the case of finite-range, frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show that a property of the ground state space is sufficient to obtain such a bound. We furthermore show that such a condition is necessary and equivalent to a constant spectral gap. Thanks to this equivalence, we can prove that for gapless models in any dimension, the spectral gap on regions of diameter $n$ is at most $o\left(\frac{\log(n)^{2+\epsilon}}{n}\right)$ for any positive $\epsilon$.